So if \(k=k^*\) , we get that \(k(xyz)=-k(xyz)\)  for all \(x,y,z\in S\) . This implies that \(k(xyz)=0\) . This completes the proof of Lemma REF .

In the following proposition we extend the results obtained by Ajebbar and Elqorachi [1]} on semigroups generated by their squares and by Ebanks [2]} on monoids to general semigroups.
